1hint : 1.use integration by parts
2.use reccurence relation ;)
y did u give xn-1 lol..cu have simply given xn as n→∞
k=\lim_{n\to\infty}\frac{(2n+1)\int_{0}^{1}x^{n-1}\sin\left(\frac{\pi}{2}x\right)dx}{(n+1)^{2}\int_{0}^{1}x^{n-1}\cos\left(\frac{\pi}{2}x\right)dx}\ \ (n=1,\ 2,\ \cdots).
find k
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7 Answers
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1ok...
\int_{0}^{1}{x^n\sin \frac{\pi x}{2}}=0+\frac{2n}{\pi}\int_{0}^{1}{x^{n-1}\cos \frac{\pi x}{2}} \\ I(n)=I'(n-1)\frac{2n}{\pi}\\ \lim_{n\rightarrow \infty}I(n)=\lim_{n\rightarrow \infty}I'(n)\frac{2n}{\pi}
now put it in ur expression
it comes as
n^2(2+1/n).2/n^2(1+1/n)^2.pi=4/pi ???